$\mathcal{A}$-Schemes and Zariski-Riemann Spaces

Abstract

In this paper, we will investigate further properties of $\mathcal{A}$-schemes introduced in [Tak]. The category of $\mathcal{A}$-schemes possesses many properties of the category of coherent schemes, and in addition, it is co-complete and complete. There is the universal compactification, namely, the Zariski–Riemann space in the category of $\mathcal{A}$-schemes. We compare it with the classical Zariski–Riemann space, and characterize the latter by a left adjoint.